Is there a selection tool that will select everything towards the cursor?
I don't think there exists any selection tool that uses the 3d cursor in this manner but there are a few that work in a similar way.

There is Select > Side of Active.


  Selects all data on the mesh in a single axis


If you select at least a vertex or an edge, it will select all the faces to the positive (right) or negative (left) axis mode of that selection based on the axis you choose in the operator settings. You can also of course just invert the selection to get the other side.

There is also Select > Loop Inner-Region.


  Converts a closed selection of edges to the region of faces it encloses


I would say this is almost the opposite of select side of active. Depending on where you select, it will select all faces to the side that has the lesser amount of faces to the end of the mesh.